Begin by preheating your oven to 350°F (175°C). Prepare a baking sheet by lining it with parchment paper to prevent sticking.
In a large mixing bowl, combine softened unsalted butter, granulated sugar, and packed brown sugar. Using a hand mixer or stand mixer, beat the mixture until it becomes creamy and well-blended, about 2-3 minutes.
Next, add the pumpkin puree, large egg, and vanilla extract to the butter-sugar mixture. Beat until all the ingredients are fully incorporated and smooth.
In a separate bowl, whisk together the all-purpose flour, baking soda, baking powder, ground cinnamon, ground nutmeg, ground ginger, and salt until well mixed.
Gradually add the dry mixture to the wet ingredients, mixing gently until just combined. Avoid overmixing to ensure your cookies remain tender.
Using a tablespoon or cookie scoop, drop spoonfuls of cookie dough onto the prepared baking sheet, spacing them about 2 inches apart. Gently flatten each cookie with the back of a spoon for even baking.
Bake the cookies in the preheated oven for 12-15 minutes, or until the edges are lightly golden brown. The centers may appear soft but will firm up as they cool.
While the cookies are cooling, prepare the creamy frosting. In a medium bowl, beat the softened cream cheese with an electric mixer until it is smooth and free of lumps.
Gradually add the powdered sugar and milk to the cream cheese. Mix until the frosting is creamy and spreadable, adjusting the consistency with more milk or powdered sugar as necessary.
Once the cookies are completely cooled, generously spread a dollop of the cream cheese frosting on top of each cookie using a knife or spatula.
For an extra flourish, finish with a light sprinkle of cinnamon or pumpkin spice atop the frosting for added flavor and visual appeal.
Notes
Arrange the frosted cookies on a festive platter and garnish with a few whole spices or mini pumpkins for an autumn-themed display.